Spectroscopists frequently use derivatives of the spectrum to extract
information. Differentiation is not really a stable
mathematical process, so the rule is that differentiating a noisy signal
will amplify the noise.

Smoothing a derivative is not quite the right thing
to do. The right thing is to compute the most probable derivative from the
infinite ensemble of noisy derivatives that noisy samples from the same
process could produce. That is what we do, using a process known as Bayesian
analysis.

Shown in Fig 1 is the second derivative of a rather
noisy signal. Note the complete absence of noise in the result. We are
not removing the noise, but estimating what the derivative would look
like if there were no noise.

Not everyone needs a Bayesian differentiator, but if you
do, you really need ours. It will handle derivatives of all orders.